SOLUTION: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥
3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞
and 𝑟 (≠ 0) are real numbers, ex
Algebra.Com
Question 1205171: If 𝛼, 𝛽, 𝛾 (where 𝛼, 𝛽, 𝛾 ≠ 0) are the roots of the equation 𝑥
3 + 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 = 0, where 𝑝, 𝑞
and 𝑟 (≠ 0) are real numbers, express the following in terms of 𝑝, 𝑞 and 𝑟:
𝛼^2𝛽^2 + 𝛽^2𝛾^2 + 𝛼^2𝛾^2
Found 2 solutions by Bogz, ikleyn:
Answer by Bogz(13) (Show Source): You can put this solution on YOUR website!
If , , are the roots of , then we know that
,
, and
.
The 2nd equation above gives
.
But the left side of the preceding equation expands to
.
Hence, .
Therefore,
Answer by ikleyn(52778) (Show Source): You can put this solution on YOUR website!
.
In this problem, the given restrictions 𝛼, 𝛽, 𝛾 ≠ 0 and 𝑟 ≠ 0 are not necessary and do not matter.
The solution, given by the other tutor, works nicely without these restrictions, too.
My conclusion: So, these restrictions can be omitted.
They are not necessary.
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